This course provides an introduction to the linear functional analysis and some ele-ments of nonlinear functional analysis. It includes the theory of metric and normed spaces (as much as it necessary), Hahn–Banah theory, theory of compact and linear operators, Fredholm theory, and spectral theory of linear operators as well as derivatives of nonlinear operators. The course contains some example of applications of functional analysis to PDE, in particular to elasticity theory and fluid mechanics.
Lectures and assignments.
It closely related to Fourier, complex, harmonic, and real analysis courses as well as to partial differential equations and mathematical physics.
Real and complex analysis including basic knowledge of metric spaces, PDE.
1.1 Definitions: metric space, energy spaces, examples
1.2 Complete, compact and separable metric spaces
2.1 Linear spaces
2.2 Normed spaces
2.3 Inner product
2.4 Banach spaces
2.5 Hilbert spaces
2.6 Lebesgue and Sobolev spaces
3.1 Linear operators. Continuity and boundedness
3.2 Spaces of linear operators
3.3 Inverse operators
4.1 Hahn–Banach theorem and its consequences
4.2 Dual spaces and operators
4.3 Projections and complementary subspaces
4.4 Weak and weak-* convergence
5.1 The Adjoint of an operator
5.2 Normal, Self-adjoint and Unitary operators
6.1 Compact sets in normed spaces
6.2 Linear completely continuous operators
8.1 Eigenvalues and eigenvectors of linear operators
8.2 The resolvent set and the spectrum of a linear operator
8.3 Spectral decomposition of self-adjoint operators
9.1 Fréchet and Gȃteaux derivatives
9.2 Lyapunov–Schmidt method
9.3 Critical point of a functional
9.4 Contracting mappings
9.5 Newton iteration process
9.6 Some applications to PDE
Linear Functional Analysis by B. Rynne and M. Youngson, Springer, 2nd edition, 2007.